Logk

Logarithm base k, written log_k(x), is the inverse function of the exponential function k^x. It is defined for positive arguments x > 0 and for bases k > 0 with k ≠ 1. In particular, log_k(x) satisfies k^{log_k(x)} = x and log_k(k^y) = y, with log_k(1) = 0.

Key identities include the change of base formula log_k(x) = ln(x) / ln(k) = log_a(x) / log_a(k) for any a > 0, a ≠ 1. It is also additive over multiplication and scales with powers: log_k(xy) = log_k(x) + log_k(y) for x,y > 0, and log_k(x^r) = r log_k(x) for any real r. The domain is x > 0 and the range is all real numbers. The function is increasing when k > 1 and decreasing when 0 < k < 1. As x approaches 0^+, log_k(x) tends to -∞ if k > 1 and to +∞ if 0 < k < 1; as x → ∞, log_k(x) tends to ∞ for k > 1 and to -∞ for 0 < k < 1.

Common examples: log_2(8) = 3, log_10(100) = 2, log_e(x) is the natural logarithm. Inverse to exponentials, log_k converts multiplicative scales to additive scales, a property widely used in science, computer science, and information theory.